Mechanistic slumber vs. statistical insomnia: The early phase of Boltzmann’s H-theorem (1868-1877)

In 1872 Ludwig Boltzmann published a lengthy memoir containing two fundamental results: an integro-differential equation describing the time evolution of an ideal gas (Boltzmann equation) and a mathematical argument proving that such evolution must reach, sooner or later, the state of equilibrium described by the Maxwell distribution. The latter achievement, later called the H-theorem, expressed in mechanistic language the essence of the second law of thermodynamics according to which thermodynamic systems tend irreversibly toward a final state. Five years later, in 1877, Boltzmann tackled the equilibrium problem from a completely different angle and emphasized that the evolution toward the Maxwell distribution is a matter of probability: the equilibrium is reached and maintained because is the overwhelmingly most probable state. How Boltzmann moved from the kinetic language of 1872 to the overtly statistical terminology of 1877 has been the subject of a long debate among the specialists.